Without further ado, nodal analysis is one of the many methods in solving and finding a specific value of a parameter in electronic circuit analysis. The aim of using nodal analysis is to determine the voltage in each node that’s relative to the reference node, which is the ground GND where voltage is equal to 0. This means that all the other nodes present in the circuit are referred to as the non-reference nodes; the ones that has voltage you are trying to solve for. Depends, of course, if you do need the voltage present in them or not.
Let’s do a quick recap about the parts of an electronic circuit; A node is a point of connection between two or more branches. A branch represents a single element such as a voltage source, or a resistor, etc. And a loop is any closed path in a circuit.
Here are the steps on how to determine node voltages:
Determine the nodes of the circuit, and then select a node as the reference node (ground GND). Then assign the non-reference nodes to voltages V1, V2, Vx, or whatever you feel comfortable with.
Apply KCL (Kirchhoff’s Current Law) to each of the non-reference nodes. Use Ohm’s law (V=IR) to express the branch currents in terms of node voltages. I use the shortcut method, though the same principles are still applied.
Solve the resulting simultaneous equations formed from the non-reference nodes to obtain the unknown node voltages.
Always remember that the number of equations formed should be equal to the number of unknowns. So, taking this simple circuit as an example:
This sample circuit has an AC voltage source, a current source, a resistor, an inductor, and a capacitor. We can also observe that there are four nodes present in this circuit. Assuming that we need to find the voltage across the capacitor, what node will we choose as our reference node GND that will make the problem easier? Note that the lesser the unknowns, the easier the problem will be. It’s like the unknowns determine the difficulty of the problem.
So, there are four nodes. If we were to select the top-left node as the GND, V1 as the top-mid node, V2 as the top-right node, and V3 as the bottom node, it would mean that we can get the voltage across the capacitor with V1 – V3 where V3 = -Vs (voltage source, since the negative terminal of the voltage source is connected to node V3). This would be a fine option but it’s kind of — maybe, unethical — to have the GND connected to the positive terminal of the voltage source.
If we were to select the top-mid node as the GND, V1 as the top-right, and the same position for V2 and V3, then a supernode (formed by enclosing a voltage source, either dependent or independent, connected between two non-reference nodes and any elements connected in parallel with it) would be present which adds more difficulty in solving the circuit. Same situation goes if the top-right node is the GND.
But if we were to select the bottom node as the GND, and V1, V2, and V3 as the top-left, top-mid, and top-right nodes, respectively, then we can get the voltage across the capacitor with V2 – GND = V2 – 0 = V2 (since current flows from a higher potential to a lower potential in a resistor), while V1 = Vs (since the positive terminal of the voltage source is connected to node V1). That makes two unknowns (V2 and V3), though we only need to solve for V2. And it’s more ethical compared to having the GND on top of the circuit.
Now to label the nodes and elements, assuming that all the elements’ respective values are given and have been converted to their equal impedances already. If you want to know how to solve for the impedance in a resistor, inductor, and a capacitor, click here.
There we go! So, what we are trying to find is the voltage across the capacitor Z3 and we have discussed earlier that Vz3 = V2, since Vz3 = V2 – GND and that the voltage at the GND is equal to 0. Now to form the equation at node V2 using the shortcut method. Brace yourself for I am about to make up names that i’ll be using to better explain the shortcut method.
@node V2:
V2( ) = 0
To form the equation at node V2, you must locate V2 (imagine V2 as a person or an animal or any object) and look around its surroundings. And I mean the lines or pathways that are connected to it. In this case, there are three pathways connected to V2 (path to V1, to V3, and to GND). And each path is connected to an element (resistor, inductor, and capacitor), or what I will be naming as a bridge. And this is how you start the equation:
V2( 1/bridge1 + 1/bridge2 + 1/bridge3) = 0
..or..
V2( 1/z1 + 1/z2 + 1/z3 ) = 0
It’s like connecting impedances in parallel. Continuing to the path across one of the bridges, you’ll encounter another node (it could either be the GND or not), or what i’ll be naming as your neighbour. This is how neighbours are treated in the equation:
V2( 1/z1 + 1/z2 + 1/z3 ) – neighbour1/bridge1 – neighbour2/bridge2 – neighbour3/bridge3 = 0
..or..
V2( 1/z1 + 1/z2 + 1/z3 ) – V1/z1 – V3/z2 – GND/z3 = 0
..and since V1 = Vs, and GND = 0..
V2( 1/z1 + 1/z2 + 1/z3 ) – Vs/z1 – V3/z2 – 0 = 0
..and since Vs/z1 is a constant, we transpose it to the other side..
V2( 1/z1 + 1/z2 + 1/z3 ) – V3/z2 = Vs/z1
And that’s it for the first equation, with V2 and V3 as unknowns. As I mentioned earlier, the number of unknowns should be equal to the number of equations. So, we are going to need one more equation, and that’d be the equation at node V3.
@node V3:
V3( ) = 0
Following the same procedures with the bridges and neighbours, we get–
V3( 1/z2 ) – V2/z2 = 0
As you can see, node V3 is connected to a current source. A current source in nodal analysis is considered as a constant unless it’s a dependent source. When the current source’s direction is away from the node, you add the current to the equation. If its direction is towards the node, you subtract the current to the equation. And since the current source in the sample circuit is directed towards node V3, it goes like this:
V3( 1/z2 ) – V2/z2 – Is = 0
..and since Is is already a given constant, we transpose it to the other side..
Vs( 1/z2 ) – V2/z2 = Is
And there’s your second equation. Now you can solve for Vz3 by fusing the two equations using matrices, substitution, elimination, or whatever method you know. Though our professor requires us to use the matrix method.
Learning:
This week we have learned to use another method, called nodal analysis in solving some values of the elements present in the circuit. The first process we need to do is to locate a reference node, a node where most of the elements are connected, second is to assign voltages across each non-reference nodes. Third is to use KCL on each non-reference nodes. Lastly in order to solve for the voltages on each nodes, we are required to use the matrix and do the math. :D
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